A107131 → (θ+1)²

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(4 + 2\,x + 12\,n - 3\,n x + 13\,n^{2} - 13\,n^{2} x + n^{2} x^{2} + 6\,n^{3} - 10\,n^{3} x + 2\,n^{3} x^{2} + n^{4} - 2\,n^{4} x + n^{4} x^{2}\right)\,P_{n}(x) = \left(8 + 4\,x + 12\,n + 22\,n x - 6\,n x^{2} + n x^{3} - 4\,n^{2} + 59\,n^{2} x - 46\,n^{2} x^{2} + 4\,n^{2} x^{3} - 12\,n^{3} + 48\,n^{3} x - 42\,n^{3} x^{2} + 8\,n^{3} x^{3} - 4\,n^{4} + 11\,n^{4} x - 10\,n^{4} x^{2} + 3\,n^{4} x^{3}\right)\,P_{n-1}(x) + \left(16\,x - 28\,x^{2} + 10\,x^{3} - x^{4} + 40\,n x - 98\,n x^{2} + 38\,n x^{3} - 4\,n x^{4} + 32\,n^{2} x - 80\,n^{2} x^{2} + 42\,n^{2} x^{3} - 6\,n^{2} x^{4} + 8\,n^{3} x - 18\,n^{3} x^{2} + 12\,n^{3} x^{3} - 2\,n^{3} x^{4}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	0	4														4
2	0	4	9													13
3	0	0	27	16												43
4	0	0	18	96	25											139
5	0	0	0	160	250	36										446
6	0	0	0	80	750	540	49									1419
7	0	0	0	0	875	2520	1029	64								4488
8	0	0	0	0	350	5040	6860	1792	81							14123
9	0	0	0	0	0	4536	20580	16128	2916	100						44260
10	0	0	0	0	0	1512	30870	67200	34020	4500	121					138223
11	0	0	0	0	0	0	22638	147840	187110	66000	6655	144				430387
12	0	0	0	0	0	0	6468	177408	561330	462000	119790	9504	169			1336669
13	0	0	0	0	0	0	0	109824	972972	1801800	1038180	205920	13182	196		4142074
14	0	0	0	0	0	0	0	27456	972972	4204200	5087082	2162160	338338	17836	225	12810269

Technical details

Node id: n-7ae5643b6f3658cb · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-d4c175ac402b3990

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1;
  1: 0, 4;
  2: 0, 4,  9;
  3: 0, 0, 27,  16;
  4: 0, 0, 18,  96,  25;
  5: 0, 0,  0, 160, 250,   36;
  6: 0, 0,  0,  80, 750,  540,    49;
  7: 0, 0,  0,   0, 875, 2520,  1029,    64;
  8: 0, 0,  0,   0, 350, 5040,  6860,  1792,   81;
  9: 0, 0,  0,   0,   0, 4536, 20580, 16128, 2916, 100;

%F Formula

The row polynomials satisfy (n^4+6*n^3+13*n^2+12*n+4-(2*n^4+10*n^3+13*n^2+3*n-2)*x+(n^4+2*n^3+n^2)*x^2)*P(n,x) = -(4*n^4+12*n^3+4*n^2-12*n-8-(11*n^4+48*n^3+59*n^2+22*n+4)*x+(10*n^4+42*n^3+46*n^2+6*n)*x^2-(3*n^4+8*n^3+4*n^2+n)*x^3)*P(n-1,x) + ((8*n^3+32*n^2+40*n+16)*x-(18*n^3+80*n^2+98*n+28)*x^2+(12*n^3+42*n^2+38*n+10)*x^3-(2*n^3+6*n^2+4*n+1)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^4 + 12*n^3 + 4*n^2 - 12*n - 8 - (11*n^4 + 48*n^3 + 59*n^2 + 22*n + 4)*x + (10*n^4 + 42*n^3 + 46*n^2 + 6*n)*x^2 - (3*n^4 + 8*n^3 + 4*n^2 + n)*x^3)*P[n - 1, x] + ((8*n^3 + 32*n^2 + 40*n + 16)*x - (18*n^3 + 80*n^2 + 98*n + 28)*x^2 + (12*n^3 + 42*n^2 + 38*n + 10)*x^3 - (2*n^3 + 6*n^2 + 4*n + 1)*x^4)*D[P[n - 1, x], x])/(n^4 + 6*n^3 + 13*n^2 + 12*n + 4 - (2*n^4 + 10*n^3 + 13*n^2 + 3*n - 2)*x + (n^4 + 2*n^3 + n^2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(4*n^4 + 12*n^3 + 4*n^2 - 12*n - 8 - (11*n^4 + 48*n^3 + 59*n^2 + 22*n + 4)*x + (10*n^4 + 42*n^3 + 46*n^2 + 6*n)*x^2 - (3*n^4 + 8*n^3 + 4*n^2 + n)*x^3)*v[i - 1] + ((8*n^3 + 32*n^2 + 40*n + 16)*x - (18*n^3 + 80*n^2 + 98*n + 28)*x^2 + (12*n^3 + 42*n^2 + 38*n + 10)*x^3 - (2*n^3 + 6*n^2 + 4*n + 1)*x^4)*deriv(v[i - 1], x))/(n^4 + 6*n^3 + 13*n^2 + 12*n + 4 - (2*n^4 + 10*n^3 + 13*n^2 + 3*n - 2)*x + (n^4 + 2*n^3 + n^2)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from A Motzkin related triangle.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.